For the sake of simplicity, consider a steady-state free-surface problem, with the force on the free surface equal to zero. (This formulation can be generalized to a nonzero , and can also be extended to moving interfaces and time-dependent problems.)

In all directions, boundary conditions are expressed as

(15–13)

and in the normal direction, they are expressed as

(15–14)

or (for time-dependent problems)

(15–15)

is the displacement of the node:

(15–16)

where is the director at node .

The fact that and are both unknowns in Equation 15–14 and Equation 15–15 makes free-surface problems nonlinear. Ansys Polyflow will therefore need several iterations to converge, starting from an initial guess. In most cases, these iterations are of the Newton- Raphson type (that is, velocity and position variables are updated at the same time), but it is also possible to use a decoupled scheme between velocity and position variables, as described at the end of Convergence Strategies. In the decoupled case, all kinematic conditions, as well as all remeshing methods, are decoupled from the flow problem.